Computational Methods for Partial Differential Equations

Transcription

1 Computtionl Methods for Prtil Differentil Equtions Mnolis Georgoulis Deprtment of Mthemtics University of Leicester e-mil: Jnury 9

2 Contents Introduction to Prtil Differentil Equtions Introduction Solution of PDEs 3 Clssifiction of PDEs 3 4 First order liner PDEs 5 5 Second order liner PDEs 8 6 The Cuchy problem nd well-posedness of PDEs 5 Problems of Mthemticl Physics 9 The Lplce eqution 9 Seprtion of vribles Fourier series 4 3 Bck to the Lplce problem 9 The het eqution nd the initil/boundry vlue problem 3 Seprtion of vribles 3 3 The wve eqution nd the initil/boundry vlue problem 33 3 Seprtion of vribles 33 3 Divided Differences 37 3 Divided Differences 37 3 Divided differences for first derivtives 37 3 Divided differences for higher derivtives 4 33 Comments nd further reding 4 3 Difference methods for two-point boundry vlue problems 4 3 Error nlysis 44 4 Finite Difference Methods for Prbolic Problems 46 4 Explicit Euler method 46 4 Error nlysis of the explicit Euler method 5 4 Stbility Anlysis 53 4 Implicit Methods 55 4 Implicit Euler Method 55 4 The Crnk-Nicolson Method Extensions to problems with non-constnt coefficients 6 43 Explicit methods 6 43 Implicit methods 6 5 Finite Difference Methods for Elliptic Problems 64 5 The five-point scheme 64 5 Error nlysis Finite difference methods for generl elliptic problems 69 6 Finite Difference Methods for Hyperbolic Problems 7 6 The CFL condition 73 6 The upwind method 76 6 Error nlysis 8 6 Stbility nlysis 8 63 The Lx-Wendroff method 83 i

3 7 The Finite Element Method 87 7 Introduction 87 7 Wek derivtives The two-point boundry vlue problem in wek form 9 74 The finite element method for the two-point boundry vlue problem 9 75 FEM in two nd three dimensions Clcultion of the FEM system for simple problem 77 FEM for prbolic problems 4 ii

4 Chpter Introduction to Prtil Differentil Equtions Introduction A prtil differentil eqution (PDE) is n eqution involving n unknown function of two or more vribles with some of its prtil derivtives PDEs re of fundmentl importnce in pplied mthemtics nd physics, nd hve recently shown to be useful in s vried disciplines s finncil modelling nd modelling of biologicl systems More specificlly, we hve the following definition Definition Let Ω R d be n open subset of R d (clled the domin of definition), for d > positive integer (clled the dimension), nd denote by x = (x, x,,x d ) vector in Ω Let (unknown) function u : Ω R whose prtil derivtives up to order k (for k positive integer) u, u,, u x x, uk x k x d, u, uk x k x, u x,, uk x k, d,, u x, d u k x k,, x u x x,, u k x d x k d u x x d, exist A prtil differentil eqution of order k in Ω in d dimensions is n eqution of the form: where F is given function F(x, u, u x, u x,, u x d,, u k x d x k d Exmple Let x = (x, x ) R nd function u : R R The eqution is PDE of st order on R in dimensions Exmple 3 Let u : [, ] 3 R The eqution u x =, u x + u y + u z = 3x3, ) =, () is PDE of nd order on [, ] 3 in 3 dimensions (This is n instnce of the so-clled Poisson eqution) Solution Indeed, this is in ccordnce with Definition with d = 3, x = x, x = y, x 3 = z nd Ω = R 3

5 Exmple 4 Let u : R 3 R The eqution u t = u x + u y, is PDE of nd order on R 3 in 3 dimensions (This is n instnce of the so-clled het eqution) Exmple 5 Let u : R R, with u = u(t, x) The eqution u t + x σ u x + rx u ru =, x is PDE of nd order on R in dimensions (This is the so-clled Blck-Scholes eqution) Exmple 6 Let u : R R The eqution u u ( x y u ) =, x y is PDE of nd order on R in dimensions (This is the so-clled Monge-Ampère eqution) We shll be mostly interested in PDEs in two nd three dimensions (s these re the ones most often ppering in prcticl pplictions), nd we shll confine the nottion to these cses using (x, y) nd (t, x) or (x, y, z) nd (t, x, y) to describe two- nd three-dimensionl vectors respectively (when the nottion t is used for n independent vrible, this vrible should lmost lwys describing time ) Nevertheless, mny properties nd ides described below pply lso to the generl cse of d-dimensions for d > 3 Also, to simplify the nottion, we shll often resort to the more compct nottion u x, u y, u xx, u xy, etc, to signify prtil derivtives u x, u y, u x, u x y, etc, respectively Solution of PDEs Studying nd solving PDEs hs been one of the centrl res of reserch in pplied mthemtics during the lst two centuries Definition 7 Consider the nottion of Definition We cll the generl solution of the PDE (), the fmily of functions u : Ω R d R tht hs continuous prtil derivtives up to (nd including) order k nd tht stisfies () Exmple 8 We wnt to find the generl solution of the PDE in R : Integrting with respect to x, we get u x = x 3 u(x, x ) = x4 4 + f(x ), () for ny differentible function f : R R Indeed, if we differentite this solution with respect to x, we get bck the PDE It is lso not hrd to see tht if u is solution of the PDE then it hs to be of the form () s this follows from the Fundmentl Theorem of Clculus (Why?) Solving PDEs is often fr more tricky pursuit thn the previous exmple seems to indicte Let us try to see why Consider, for exmple, the PDE u + u = x x It is not too hrd to guess tht ny constnt function u stisfies this PDE However, there re more functions tht stisfy this PDE tht just the constnt ones It is cler tht n integrtion will not be of ny help here nd more elborte methods need to be introduced Moreover, s we shll see below, there is no method of solving PDEs tht works in generl: different methods work for different fmilies of PDEs Therefore, it is importnt to identify such fmilies of PDEs tht dmit similr properties nd, subsequently to describe prticulr methods of solving PDEs from ech such fmily This will be the content of the rest of this chpter, where we shll clssify PDEs in vrious fmilies nd present some of their properties

6 3 Clssifiction of PDEs To study PDEs it is often useful to clssify them into vrious fmilies, since PDEs belonging to prticulr fmilies cn be chrcterised by similr behviour nd properties There re mny nd vried clssifictions for PDEs Perhps the most widely ccepted nd generlly useful clssifiction is the distinction between liner nd non-liner PDEs In prticulr, we hve the following definition Definition 9 If the PDE () cn be written in the form (x)u + b (x)u x + b (x)u x + + b d (x)u xd + c (x)u xx + + c (x)u xx + + c d (x)u xd x d + = f(x), (3) ie, if the coefficients of the unknown function u nd of ll its derivtives depend only on the independent vribles x = (x, x,, x d ), then it is clled liner PDE If it is not possible to write () in the form (3), then it is clled nonliner PDE Exmple The PDEs in Exmples, 3, 4, nd 5 re liner PDEs Indeed, the PDE in Exmple, cn be written in the form (3) with (x) =, b (x) =, f(x) = nd ll the other coefficients of the derivtives equl to zero Similrly, for Exmple 3, we hve f(x) = 3x 3, the coefficients of the second derivtives u xx, u yy nd u zz re equl to nd ll the other coefficients re zero Also, for Exmple 4, f(x) =, the coefficients of u t, u xx nd u yy re equl to nd ll the other coefficients re zero Finlly, for Exmple 5, f(x) =, the coefficients of u t, u xx, u x, nd u depend only on the independent vrible x nd do not depend of u Exmple The PDE in Exmple 6 nonliner PDE This is cler, since the coefficient of u xx is equl to u yy (or to put it differently: the coefficient of u yy is equl to u xx ) nd the coefficient of u xy is equl to u xy, ie, the coefficients of t lest one of the prtil derivtives contin u or its derivtives Exmple The inviscid Burgers eqution u t + uu x =, for n unknown function u = u(t, x) is nonliner PDE The fmily of nonliner PDEs cn be further subdivided into smller fmilies of PDEs In prticulr we hve the following definition Definition 3 Consider nonliner PDE of order k with unknown solution u If the coefficients of the k order prtil derivtives of u re functions of the independent vribles x = (x, x,, x d ) only, then this is clled semiliner PDE If the coefficients of the k order prtil derivtives of u re functions of the independent vribles x = (x, x,, x d ) nd/or of prtil derivtives of u of order t most k (including u itself), then this is clled qusiliner PDE If (nonliner) PDE is not qusiliner, then it is clled fully nonliner Clerly semiliner PDE is lso qusiliner PDE Exmple 4 We give some exmples of nonliner PDEs long with their clssifictions The rection-diffusion eqution is semiliner PDE u t = u xx + u, The inviscid Burgers eqution u t + uu x =, is qusiliner PDE nd it is not semiliner PDE 3

7 The Korteweg-de Vries (KdV) eqution u t + uu x + u xxx =, is semiliner PDE The Monge-Ampère eqution is fully nonliner PDE u xx u yy (u xy ) =, The bove clssifiction of PDEs into liner, semiliner, qusiliner, nd fully nonliner is, roughly speking, clssifiction of incresing difficulty in terms of studying nd solving PDEs Indeed, the mthemticl theory of liner PDEs is now well understood On the other hnd, less is known bout semiliner PDEs nd qusiliner PDEs, nd even less bout fully nonliner PDEs Problem Consider the following PDEs: the bihrmonic eqution: u xxxx u xxyy + u yyyy = f(x, y), with u = u(x, y); the sine-gordon eqution: u tt u xx + sinu =, with u = u(t, x); 3 the porous medium eqution: u t = (u n u x ) x, with u = u(t, x) nd n positive integer Wht is the order of ech PDE? Clssify the PDEs 4

8 4 First order liner PDEs We begin our study of liner PDEs with the cse of first order liner PDEs To simplify the discussion,we shll only consider equtions in dimensions, ie, for d = ; the cse of three or more dimensions cn be treted in completely nlogous fshion We begin with n exmple Exmple 5 We consider the PDE in R : u x + u y = (4) To find its generl solution, we perform the following trnsformtion of coordintes (lso known s chnge of vribles in Clculus): we consider new vribles (ξ, η) R defined by the trnsformtion of coordintes (x, y) (ξ, η),where ξ(x, y) = x + y nd η(x, y) = y x We cn lso clculte the inverse trnsformtion of coordintes by solving with respect to x nd y, obtining (ξ, η) (x, y), x = (ξ η) nd y = (ξ + η) (5) We write the PDE (4) in the new coordintes, using the chin rule from Clculus Setting v(ξ, η) = u(x(ξ, η), y(ξ, η)) we hve, respectively: giving Putting these bck to the PDE (4), we deduce u x = v ξ ξ x + v η η x, u y = v ξ ξ y + v η η y, u x = v ξ v η, u y = v ξ + v η Integrting this eqution with respect to ξ, we rrive to = u x + u y = v ξ v η + v ξ + v η = v ξ or v ξ = (6) v(ξ, η) = f(η), for ny differentible function of one vrible f : R R Using now the inverse trnsformtion of coordintes (5), we conclude tht the generl solution of the PDE (4) is given by: u(x, y) = v(ξ(x, y), η(x, y)) = f(η(x, y)) = f(y x) We discuss some big ides tht re present in the previous exmple The chnge of vribles (x, y) (ξ, η) is essentilly clockwise rottion of the xes by n ngle π 4 (there is lso stretching of size tking plce with this chnge of vribles, but this is not relly relevnt to our discussion) Once the rottion is done, the PDE tkes the simpler form (6), which cn be interpreted geometriclly s: v is constnt with respect to the vrible ξ, or in other words, the solution u is constnt when x + y = c, for ny constnt c R This mens tht the solution remins constnt s we move long stright lines of the form y = x + c Hence, if the vlue of the solution u t one point (x, y ), sy, on the plne is known, then the vlue of u long the stright line of slope π 4 tht psses through (x, y ) is lso known (ie, it is the sme vlue)! In other words, the stright lines of the form y = x + c chrcterise the solution of the PDE bove; such curves re clled chrcteristic curves of PDE, s we shll see below Next, we shll incorporte these ides into the cse of the generl first order liner PDE The generl form of st order liner PDE in dimensions cn be written s: (x, y)u x + b(x, y)u y + c(x, y)u = g(x, y), for (x, y) Ω R, (7) where, b, c, g re functions of the independent vribles x nd y only We lso ssume tht, b hve continuous first prtil derivtives, nd tht they do not vnish simultneously t ny point of the domin of definition Ω Finlly,we ssume tht the solution u of the PDE (7) hs continuous first prtil derivtives 5

9 Consider trnsformtion of coordintes of R : (x, y) (ξ, η), with ξ = ξ(x, y) nd η = η(x, y), which is ssumed to be smooth (tht is, the functions ξ(x, y) nd η(x, y) hve ll derivtives with respect to x nd y well-defined) nd non-singulr, ie, its Jcobin (ξ, η) (x, y) := ξ x ξ y = ξx η η x η y ξ y η x, (8) y in Ω; (this requirement ensures tht the chnge of vribles is meningful, in the sense tht it is one-to-one nd onto) We lso denote by x = x(ξ, η) nd y = y(ξ, η) the inverse trnsformtion, s it will be useful below We write the PDE (7) in the new coordintes, using the chin rule Setting v(ξ, η) = u(x(ξ, η), y(ξ, η)) we hve, respectively: u x = v ξ ξ x + v η η x, u y = v ξ ξ y + v η η y, giving (ξ x + bξ y )v ξ + (η x + bη y )v η + cv = g(x(ξ, η), y(ξ, η)), (9) fter substitution into (7) To simplify the bove eqution, we require tht the function η(x, y) is such tht η x + bη y = ; () if this is the cse then (9) becomes n ordinry differentil eqution with respect to the independent vrible ξ, whose solution cn be found by stndrd seprtion of vribles The eqution () is slightly simpler PDE of first order thn the originl PDE To find the required η we re seek to construct curves such tht η(x, y) = const for ny constnt; these re clled the chrcteristic curves of the PDE (compre this with the stright lines of the exmple bove) Differentiting this eqution with respect to x, we get = d const dx = dη(x, y) dx dx = η x dx + η dy y dx = η dy x + η y dx, where in the penultimte equlity we mde use of the chin rule for functions of two vribles; the bove equlity yields η x = dy η y dx, () ssuming, without loss of generlity, tht η y (for otherwise, we rgue s bove with the rôles of the x nd y vribles interchnged, nd we get necessrily η x from hypothesis (8)) Using () on (), we deduce the chrcteristic eqution: dy dx + b =, or dy dx = b, () ssuming, without loss of generlity tht ner the point (x, y ) (for otherwise, we hve tht necessrily b ner the point (x, y ), s, b cnnot vnish simultneously t ny point due to hypothesis, nd we cn pply the sme rgument s bove with x nd y interchnged) Eqution () is n ordinry differentil eqution of first order tht cn be solved using stndrd seprtion of vribles to give solution f(x, y) = const, sy Setting η = f(x, y) nd ξ to be ny function for which (8) holds, we cn esily see tht () holds lso Therefore, the PDE (7) cn be written s (ξ x + bξ y )v ξ + cv = g(x(ξ, η), y(ξ, η)), or v ξ + c g(x(ξ, η), y(ξ, η)) v =, (ξ x + bξ y ) (ξ x + bξ y ) which is n ordinry differentil eqution of first order with respect to ξ nd cn be solved using the (stndrd) method of multipliers to find v(ξ, η) Using the inverse trnsformtion of coordintes, we cn now find the We recll the method of multipliers to solve this ordinry differentil eqution (ODE): we write the ODE in the form v ξ + Cv = G, by dividing by (ξ x + bξ y), nd we multiply both sides by e R C(s)ds, which gives er ξ C(s)ds v ξ + CeR ξ C(s)ds v = GeR ξ C(s)ds, or (er ξ C(s)ds v) ξ = GeR ξ C(s)ds, or v = e R ξ C(s)ds Z ξ G(τ)eR τ C(s)ds dτ 6

10 solution u(x, y) from v(ξ, η) This is the so-clled method of chrcteristics in finding the solution to first order PDE Exmple 6 We use the method of chrcteristics described bove to find the generl solution to the PDE yu x xu y + yu = xy We hve = y, b = x, c = y nd g = xy To find the chrcteristic curves of this PDE, we solve the ordinry differentil eqution (), which in this cse becomes dy dx = x y or using seprtion of vribles We set ydy = xdx, or y = x + const, for xy, η(x, y) := x + y ; then we hve η(x, y) = const s required by the method described bove, ie, the chrcteristic curves of this PDE re concentric circles centred t the origin If we lso set ξ(x, y) := x, sy, we hve (ξ, η) (x, y) = ξ xη y ξ y η x = y x = y, when xy Hence the trnsformtion of coordintes (x, y) (ξ, η) is non-singulr nd smooth The inverse trnsformtion is given by η x(ξ, η) = ξ nd y(ξ, η) = ξ, if y ; η ξ, if y < giving the trnsformed PDE v ξ + y(ξ, η) y(ξ, η) + ( x(ξ, η)) v = x(ξ, η)y(ξ, η) y(ξ, η) + ( x(ξ, η)), or v ξ + v = x(ξ, η) = ξ Multiplying the lst eqution with the multiplier e R ξ ds = e ξ, we deduce ( e ξ v ) ξ = ξeξ, or v = e ξ ξ τe τ dτ = = ξ + f(η), for ny differentible function f of one vrible Hence the generl solution is given by u(x, y) = v(ξ(x, y), η(x, y)) = ξ(x, y) + f(η(x, y)) = x + f ( x + y ) Problem Use the method of chrcteristics to find the generl solution of the first order liner PDE: u x u y = 7

11 5 Second order liner PDEs An importnt clss of PDEs re the liner PDEs of nd order, which we shll be concerned in this section For simplicity, we shll consider only equtions in dimensions, ie, for d = The generl form of nd order liner PDE in dimensions cn be written s: u xx + bu xy + cu yy + du x + eu y + fu = g, for (x, y) Ω R, (3) where, b, c, d, e, f, g re functions of the independent vribles x nd y only We lso ssume tht, b, c hve continuous second prtil derivtives, nd tht they do not vnish simultneously t ny point of the domin of definition Ω Finlly, we ssume tht the solution u of the PDE (3) hs continuous second prtil derivtives We shll clssify PDEs of the form (3) in different types, depending on the sign of the discriminnt defined by D := b c, t ech point (x, y ) Ω More specificlly, we hve the following definition Definition 7 Let D = b c be the discriminnt of second order PDE of the form (3) in Ω R nd let point (x, y ) Ω If D > t the point (x, y ), the PDE is sid to be hyperbolic t (x, y ) If D = t the point (x, y ), the PDE is sid to be prbolic t (x, y ) If D < t the point (x, y ), the PDE is sid to be elliptic t (x, y ) The eqution is sid to be hyperbolic, prbolic or elliptic in the domin Ω if it is, respectively, hyperbolic, prbolic or elliptic t ll points of Ω We give some exmples Exmple 8 The so-clled wve eqution u tt u xx =, is hyperbolic in R Indeed, for this eqution we hve =, c =, nd b =, giving D = < for ll (x, y) R Exmple 9 The so-clled het eqution u t u xx =, is prbolic in R Indeed, for this eqution we hve c = nd = b =, giving D = for ll (x, y) R Exmple The so-clled Lplce eqution u xx + u yy =, is elliptic in R Indeed, for this eqution we hve = c = nd b =, giving D = < for ll (x, y) R Exmple The eqution u xx + x u yy =, is elliptic in the set {(x, y) R : x } nd prbolic in the set {(x, y) R : x = } Indeed, for this eqution we hve =, c = x nd b =, giving D = x < for ll (x, y) R such tht x nd D = when x = (This eqution sometimes referred to in the literture s Grušin eqution) Exmple The Tricomi eqution yu xx + u yy =, is elliptic in the set {(x, y) R : y > }, prbolic in the set {(x, y) R : y = }, nd hyperbolic in the set {(x, y) R : y < } Indeed, for this eqution we hve = y, c = nd b =, giving D = y < for ll (x, y) R such tht y >, D = when y = nd D = y > when y < 8

12 The bove clssifiction of nd order liner PDEs cn be very useful when studying the properties of such equtions For instnce, the next result shows tht the type of nd order liner PDE t point (ie, if the PDE is hyperbolic, prbolic or elliptic), remins unchnged if we mke smooth non-singulr trnsformtion of coordintes (lso known s chnge of vribles) on R In prticulr, we hve the following theorem Theorem 3 The sign of the discriminnt D of second order PDE of the form (3) in Ω R is invrint under smooth non-singulr trnsformtions of coordintes Proof Consider trnsformtion of coordintes of R : (x, y) (ξ, η), with ξ = ξ(x, y) nd η = η(x, y), which is ssumed to be smooth (tht is, the functions ξ(x, y) nd η(x, y) hve ll derivtives with respect to x nd y well-defined) nd non-singulr, ie, its Jcobin (ξ, η) (x, y) := ξ x ξ y = ξx η η x η y ξ y η x, (4) y in Ω We lso denote by x = x(ξ, η) nd y = y(ξ, η) the inverse trnsformtion, s it will be useful below We write the PDE (3) in the new coordintes, using the chin rule Setting v(ξ, η) = u(x(ξ, η), y(ξ, η)) we hve, respectively: u x = v ξ ξ x + v η η x, u y = v ξ ξ y + v η η y, (5) giving u xx = v ξξ ξ x + v ξη ξ x η x + v ηη η x + v ξ ξ xx + v η η xx, u yy = v ξξ ξ y + v ξηξ y η y + v ηη η y + v ξξ yy + v η η yy, (6) u xy = v ξξ ξ x ξ y + v ξη (ξ x η y + ξ y η x ) + v ηη η x η y + v ξ ξ xy + v η η xy Inserting (5) nd (6) into (3), nd fctorising ccordingly, we rrive to where the new coefficients A, B, C, D nd E re given by Av ξξ + Bv ξη + Cv ηη + Dv ξ + Ev η + fv = g, (7) A = ξ x + bξ xξ y + cξ y, B = ξ x η x + b(ξ x η y + ξ y η x ) + cξ y η y, C = η x + bη xη y + cη y, (8) D = ξ xx + bξ xy + cξ yy + dξ x + eξ y, E = η xx + bη xy + cη yy + dη x + eη y Thus the discriminnt of the PDE in new vribles (7), is given by B AC = (ξ x η x + b(ξ x η y + ξ y η x ) + cξ y η y ) (ξx + bξ x ξ y + cξy)(η x + bη x η y + cηy) ( (ξ, η) ) = = (b c)(ξ x η y ξ y η x ) = (b c) (9) (x, y) This mens tht the discriminnt B AC of (7) hs lwys the sme sign s the discriminnt b c of (3), s (ξ,η) (x,y) from the hypothesis nd, therefore, ( (ξ,η) (x,y)) > Since the discriminnt of the trnsformed PDE hs lwys the sme sign s the one of the originl PDE, the type of the PDE remins invrint Reclling now the methods of chrcteristics for the solution of first order liner PDEs presented in the previous section, we cn see the relevnce of the bove theorem: it ssures us tht pplying chnge of vribles will not lter the type of the PDE Let us now consider some specil trnsformtions for PDEs of ech type Wht we shll see is tht, given certin trnsformtion, it is possible to write (3) loclly in much simpler form, the so-clled cnonicl form 9

13 Exmple 4 Consider the wve eqution u xx y yy =, which s we sw before is hyperbolic in R Let us lso consider the trnsformtion of coordintes of R : (x, y) (ξ, η), with ξ = x + y nd η = x y It is, of course, smooth s x + y nd x y re infinite times differentible with respect to x nd y, nd it is non-singulr, s (ξ, η) (x, y) = ξ xη y ξ y η x = ( ) =, for ll (x, y) R To clculte the trnsformed eqution, we cn use the formuls (8), with =, b =, c =, d = e = f = g =, ξ x =, ξ y =, η x = nd η y = to clculte A = =, B = + =, C = =, D = E = nd, thus, the trnsformed eqution is given by 4v ξη =, or v ξη = For this cnonicl form, we cn in fct compute the generl solution of the wve eqution Indeed, integrting with respect to η, we rrive to v ξ = h(ξ), for n rbitrry continuously differentible function h Integrting now the lst equlity with respect to ξ, we deduce ξ v = h(s)ds + G(η) If we set F(ξ) := ξ h(s)ds, to simplify the nottion, we get v(ξ, η) = F(ξ) + G(η), for n rbitrry twice continuously differentible function G, or equivlently u(x, y) = F(x + y) + G(x y), for ll twice continuously differentible functions F nd G of one vrible In the previous exmple, we found generl solution for the wve eqution using prticulr trnsformtion of coordintes Note tht the generl solution of the wve eqution involves unknown functions! (Recll tht the generl solution of n ordinry differentil eqution involves unknown constnts; this might help to drw n nlogy In the next sections we shll study some pproprite initil conditions nd boundry conditions tht will be sufficient to specify the unknown functions nd rrive to unique solutions) Problem 3 Consider the PDE: ( M )u xx + u yy = (This eqution models the potentil of the velocity field of fluid round plnr obstcle; M is clled the Mch number) Wht is the type of the bove second order liner PDE for different vlues of M? If you know wht sonic boom is, cn you see reltion to it nd the properties of the eqution bove?

14 We now investigte the following question: is it lwys possible to find trnsformtions of coordintes tht mke the generl PDE (3) simpler? In Exmple (4) we sw tht for the cse of the wve eqution it is indeed possible to reduce the wve eqution in the simpler PDE v ξη = For the generl PDE, we employ geometric rgument We seek functions ξ(x, y) nd η(x, y) for which we hve ξx + bξ xξ y + cξy = nd η x + bη xη y + cηy = ; () ie, A = C = for the coefficients of the trnsformed PDE (7) The equtions () re PDEs of first order, for which we re now seeking to construct curves such tht ξ(x, y) = const for ny constnt When (x, y) re points on curve, ie, they re such tht ξ(x, y) = const, they re dependent Hence, differentiting this eqution with respect to x, we get = d const dx = dξ(x, y) dx dx = ξ x dx + ξ dy y dx = ξ dy x + ξ y dx, where in the penultimte equlity we mde use of the chin rule for functions of two vribles; the bove equlity yields ξ x = dy ξ y dx, () ssuming, without loss of generlity, tht ξ y (for otherwise, we rgue s bove with the rôles of the x nd y vribles interchnged, nd we get necessrily ξ x from hypothesis (4)) Now, we go bck to the desired equtions (), nd we divide the first eqution by ξ y to obtin ( ξx ) ξ x + b + c =, ξ y ξ y nd, using (), we rrive to ( dy ) dy b + c =, () dx dx which is clled the chrcteristic eqution for the PDE (3) This is qudrtic eqution for dy dx, with discriminnt D = b c! The roots of the chrcteristic eqution re given by dy dx = b ± D (3) Ech of the equtions bove is first order ordinry differentil eqution tht cn be solved using stndrd seprtion of vribles to give (fmilies of) solutions f (x, y) = const nd f (x, y) = const, sy The curves defined by the equtions f (x, y) = const nd f (x, y) = const re clled the chrcteristic curves of the second order PDE Therefore, if the originl PDE (3) is hyperbolic, ie, if D >, the chrcteristic eqution hs two rel distinct roots, giving two rel distinct chrcteristics curves for the PDE If the originl PDE (3) is prbolic, thereby D =, the chrcteristic eqution hs one double root, giving one rel chrcteristic curve for the PDE Finlly, if the originl PDE (3) is elliptic, thereby D <, the chrcteristic eqution hs no rel roots, nd therefore the PDE hs no rel chrcteristic curves, but s we shll see below it hs complex chrcteristic curves The chrcteristic curves cn be thought s the nturl directions in which the PDE communictes informtion to different points in its domin of definition Ω With this sttement in mind, it is possible to see tht ech type of PDE models different phenomen nd lso dmits different properties, rendering the bove clssifiction into hyperbolic, prbolic nd elliptic PDEs of gret importnce The cse of hyperbolic PDE Now we go bck to the question of the possibility of simplifiction of the originl PDE (3), ssuming tht (3) is hyperbolic Therefore, s we hve seen bove it will hve two rel distinct chrcteristics for which the eqution (), nd thus () holds too (by observing tht ll the steps followed bove re in fct equivlences) This mens tht for every (x, y ) Ω there exists locl trnsformtion of coordintes (x, y) (ξ, η) with ξ = f (x, y) nd η = f (x, y) such tht A = C = in (7) (ie, we cn use one chrcteristic curve for ech new vrible, since both functions stisfy (), nd thus () Finlly, we check if this trnsformtion of coordintes hs non-zero Jcobin: (ξ, η) ( (x, y) = ξ ξx xη y ξ y η x = ξ y η y η ) ( x b + D == ξ y η y ξ y η y b D ) D = ξ y η y,

15 whereby the penultimte equlity follows from () Thus, we hve essentilly proven the following theorem Theorem 5 Let (3) be hyperbolic PDE Then, for every (x, y ) Ω there exists trnsformtion of coordintes (x, y) (ξ, η) in the neighbourhood of (x, y ), such tht (3) cn be written s v ξη + = g, (4) where re used to signify the terms involving u, u x, or u y This is clled the cnonicl form of hyperbolic PDE Proof The proof essentilly follows from the bove discussion: since we re ble to show tht for every (x, y ) Ω there exists locl trnsformtion of coordintes (x, y) (ξ, η) for which we hve A = C = in (7), then (7) becomes Bv ξη + Dv ξ + Ev η + fv = g Dividing now the bove eqution by B (which is not zero, s it cn be seen from (9)), (4) follows The bove theorem shows tht ech second order liner hyperbolic PDE cn be written in the (simpler) cnonicl form (4) Exmple 6 We shll clculte the chrcteristic curves of the wve eqution (Exmple (8) In this cse we hve =, b =, nd c = Thus the chrcteristic eqution reds: ( dy ) dy =, or dx dx = ±, from which we get two solutions y x = C nd y + x = C, for C, C R rbitrry constnts This yields the trnsformtion of coordintes ξ = y x nd η = y + x Compring this to (Exmple (8), we cn see tht we hve rrived to the sme trnsformtion of coordintes! Exmple 7 We shll clculte the chrcteristic curves nd the cnonicl form of the Tricomi eqution yu xx + u yy = In this cse we hve = y, b =, nd c = As we sw in Exmple (), this eqution is hyperbolic for y <, prbolic for y = nd elliptic for y > We first consider the cse y < Then the chrcteristic eqution reds: ( dy ) dy y + =, or dx dx = ±, y from which we get two solutions in implicit form 3 ( y)3/ +x = C nd 3 ( y)3/ x = C, for C, C R rbitrry constnts This yields the trnsformtion of coordintes ξ = 3 ( y)3/ + x nd η = 3 ( y)3/ x Notice tht for y = (ie, when the PDE is prbolic), we hve ξ = η, ie, the two chrcteristic curves meet, ie, we only hve one chrcteristic direction! Moreover, ξ nd η re not well defined for y >, which is gin consistent with the theory developed bove, s when y > the PDE is elliptic nd, therefore, it hs no rel chrcteristic curves! (More detils bout the lst two cses cn be found in the discussion below) Also, it is simple (but worthwhile) exercise to verify tht, with the bove chnge of vribles, the Tricomi eqution cn be written in the cnonicl form (4) when y < The cse of prbolic PDE We now ssume tht (3) is prbolic, ie, D = b c = Therefore, the eqution () hs one double root given by dy dx = b (5) which yields one fmily of chrcteristic curves, sy f (x, y) = const for which (5), nd thus () holds We set η = f (x, y) s before As fr s ξ is concerned, we now hve flexibility in its choice: the only We remind the reder how n ordinry differentil eqution is solved using seprtion of vribles: we hve dy dx = ± Z Z ydy, or ydy = ±dx, or = ±dx, or y 3 ( y)3/ = ±x + const

16 requirement is tht the Jcobin of the resulting trnsformtion of coordintes (x, y) (ξ, η) is non-zero, ie, (ξ, η) (x, y) = ξ xη y ξ y η x (6) If the bove re true, we hve C = (from the choice of ξ) in (7) Moreover, in this cse we necessrily hve tht B =, too This is becuse the PDE is prbolic, ie, D =, which from (9) nd (6) implies tht B AC = But C =, giving finlly B = lso Thus, we hve essentilly proven the following theorem Theorem 8 Let (3) be prbolic PDE Then, for every (x, y ) Ω there exists trnsformtion of coordintes (x, y) (ξ, η) in the neighbourhood of (x, y ), such tht (3) cn be written s v ξξ + = g, (7) where re used to signify the terms involving u, u x, or u y This is clled the cnonicl form of prblic PDE Proof The proof essentilly follows from the bove discussion: since we re ble to show tht for every (x, y ) Ω there exists locl trnsformtion of coordintes (x, y) (ξ, η) for which we hve B = C = in (7), then (7) becomes Av ξξ + Dv ξ + Ev η + fv = g Dividing now the bove eqution by A (which is necessrily non-zero from ssumption (6)), the result follows The bove theorem shows tht ech second order liner prbolic PDE cn be written in the (simpler) cnonicl form (7) The cse of elliptic PDE We now ssume tht (3) is elliptic, ie, D < Therefore, the eqution () hs no rel roots nd, therefore, if (3) is elliptic then it hs no rel chrcteristic curves Since complex vribles (nd the theory of nlytic functions) re beyond the scope of these notes, we shll only stte the min result for elliptic problems, without proof Theorem 9 Let (3) be n elliptic PDE Then, for every (x, y ) Ω there exists trnsformtion of coordintes (x, y) (ξ, η) in the neighbourhood of (x, y ), such tht (3) cn be written s v ξξ + v ηη + = g, (8) where re used to signify the terms involving u, u x, or u y This is clled the cnonicl form of n elliptic PDE The bove theorem shows tht ech second order liner prbolic PDE cn be written in the (simpler) cnonicl form (8) Remrk 3 Notice tht the whole discussion in this section bout liner second order PDEs will still be vlid for the cse of semiliner second order PDEs too! Indeed, since in second order semiliner PDEs the non-linerities re not present in the coefficients of the second order derivtives, the clcultions nd the theorems bove will still be vlid (s ll the clcultions bove re done to control the coefficients of the second order derivtives) Problem 4 The Blck-Scholes eqution for Europen cll option with vlue C = C(τ, s) (τ the time vrible nd s is the sset price), is given by C τ + σ s C ss + rsc s rc =, (9) 3

17 where r is positive constnt (the interest rte) Wht type of nd order liner PDE is (9) nd why? Using the following trnsformtion of coordintes of R : (τ, s) (t, x), with τ = T t σ, nd s = ex, where T is constnt (the finl time), show tht (9) cn be trnsformed into the following PDE in cnonicl form: v xx + (k )v x v t kv =, (3) where v(t, x) := C(τ(t, x), s(t, x)) = C(T t/σ, e x ), nd k := r/σ Setting now v(t, x) = e αx+βt u(t, x), for some function u = u(t, x), show tht the trnsformed eqution (3) cn be written s u t u xx =, when α = (k ), nd β = 4 (k + ), ie, the Blck-Scholes eqution cn be trnsformed into the het eqution! 4

18 6 The Cuchy problem nd well-posedness of PDEs In the previous sections, we studied the method of chrcteristics for the solution of first nd second order liner PDEs We found tht, normlly, the generl solutions of these PDEs contin unknown functions We lso gve some heuristic rguments on the importnce of chrcteristic curves in describing the properties nd the solution of PDEs In prticulr, we mentioned tht informtion trvels long chrcteristic curves, whenever these exist, ie, the solution of the PDE hs preferred direction(s) to relte its vlues from one point in spce to nother In the theory of ordinry differentil equtions, we hve seen tht the generl solution of n ODE involves unknown constnts, which cn be determined when we equip the ODE with some initil condition, eg, the ODE du(t) = 3u(t), dt hs generl solution given by u(t) = Ae 3t, for ll constnts A R If we dd the requirement tht the solution of the bove ODE must stisfy lso the initil condition u() = 5, we find tht necessrily A = 5, giving the solution u(t) = 5e 3t In this section, we shll study some pproprite corresponding conditions for PDEs, tht will be sufficient to specify the unknown functions nd rrive to unique solutions Definition 3 Consider PDE of the form (), of order k in Ω in d dimensions nd let S be (given) smooth surfce on R d Let lso n = n(x) denote the unit norml vector to the surfce S t point x = (x, x,,x d ) S Suppose tht on ny point x of the surfce S the vlues of the solution u nd of ll its directionl derivtives up to order k in the direction of n re given, ie, we re given functions f, f,,f k : S R such tht u(x) = f (x), nd u n (x) = f (x), nd u n (x) = f (x),, nd k u n k (x) = f k (x) (3) The Cuchy problem consists of finding the unknown function(s) u tht stisfy simultneously the PDE nd the conditions (3) The conditions (3) re clled the initil conditions nd the given functions f, f,,f k, will be referred to s the initil dt The degenerte cse of d = nd k =, ie, the cse of n ODE of first order with the corresponding initil condition is given bove We now consider some less trivil exmples Exmple 3 We wnt to find solution to the Cuchy problem consisting of the PDE together with the initil condition u x + u y =, (3) u(, y) = sin y (Here the surfce S in Definition 3 is implicitly given by the initil condition: we hve S = {(x, y) R : x = }, ie, the surfce S (which is now just curve s we re in R ) is the y-xis on the Crtesin plne) In Exmple 5, we used the method of chrcteristics to deduce tht the generl solution to the PDE (3) is u(x, y) = f(y x), for ll (x, y) R If we set x = we get, using the initil condition: Hence solution to the Cuchy problem is given by sin y = u(, y) = f(y) u(x, y) = sin(y x) In Figure we sketch S for this problem, long with some chrcteristic curves (which re of the form y = x + c) Notice tht S intersects ll chrcteristic curves 5

19 S chrcteristic curves Figure : Exmple 3 Sketch of the Cuchy problem Exmple 33 We wnt to find solution to the Cuchy problem consisting of the wve eqution together with the initil conditions u xx u yy =, (33) u(x, ) = sin x, nd u y (x, ) = (Agin, here the surfce S in Definition 3 is implicitly given by the initil condition: we hve S = {(x, y) R : y = }, ie, the surfce S (which is now just curve s we re in R ) is the x-xis on the Crtesin plne) In Exmple 4, we used the method of chrcteristics to deduce tht the generl solution to the PDE (33) is u(x, y) = F(x + y) + G(x y), for ll (x, y) R, for some functions F, G; thus, if we set y = we get, using the first initil condition: Differentiting the generl solution with respect to y, we get sin x = u(x, ) = F(x) + G(x) (34) u y (x, y) = F (x + y)(x + y) y + G (x y)(x y) y = F (x + y) G (x y); setting y = nd using the first initil condition, we rrive to = u y (x, ) = F (x) G (x), or F(x) G(x) = c, (35) for some constnt c R Solving the system (34) nd (35) with respect to F(x) nd G(x) (two equtions with two unknowns!), we get F(x) = (sin x + c), nd G(x) = (sin x c) Now tht we hve specified F nd G, we cn write the solution to the bove Cuchy problem u(x, y) = F(x + y) + G(x y) = (sin(x + y) + c) + (sin(x y) c) = (sin(x + y) + sin(x y)) Notice tht S intersects ll chrcteristic curves One question tht rises is whether the solutions to the Cuchy problems in the previous exmples re unique A prtil nswer to this question is given by the celebrted Cuchy-Kovlevsky Theorem Theorem 34 (The Cuchy-Kovlevsky Theorem) Consider the Cuchy problem from Definition (3) for the cse of liner PDE of the form (3) Let x be point of the initil surfce S, which is ssumed to be nlytic 3 Suppose tht S is not chrcteristic surfce t the point x Assume tht ll the coefficients of the PDE (3), the right-hnd side f, nd ll the initil dt f, f,, f k re nlytic functions on neighbourhood of the point x Then the Cuchy problem hs solution u, defined in the neighbourhood of x Moreover, the solution u is nlytic in neighbourhood of x nd it is unique in the clss of nlytic functions The proof of the bove Theorem is out of the scope of these notes; it cn be found in ny stndrd PDE theory textbook 3 An nlytic surfce which cn be described by function g(x) = const for g nlytic function An nlytic function is function tht cn be written s (bsolutely convergent) power series 6

20 Therefore, ccording to the Cuchy-Kovlevsky Theorem (under the nlyticity ssumptions), the Cuchy problem hs solution which is unique in the spce of nlytic functions Showing existence nd uniqueness of solutions to PDE problems (ie, PDEs together with some initil or boundry conditions) is, undoubtedly, tsk of prmount importnce in the theory of PDEs Indeed, once PDE model is studied, it is extremely useful to know if tht model hs solution (for otherwise, we re wsting our efforts trying to solve it) If it does hve solution, then it is very importnt to be ble to show tht the solution is lso unique (for otherwise, the sme PDE problem will produce mny different solutions, nd this is not usully nturl in mthemticl modelling) Even if PDE problem hs unique solution, this does not necessrily men tht the PDE problem is well behved By well-behved here we understnd if the PDE problem chnges slightly (eg, by ltering slightly some coefficient), then lso its solution should chnge only slightly lso In other words, well behved is to be understood s follows: smll chnges in the initil dt or the PDE itself should not result to rbitrrily lrge chnges in the behviour of the solution to the PDE problem Definition 35 A PDE problem is well-posed if the following 3 properties hold: the PDE problem hs solution the solution is unique the solution depends continuously on the PDE coefficients nd the problem dt If PDE problem is not well-posed, then we sy tht it is ill-posed The concept of well-posedness is due to Hdmrd 4 Exmple 36 The Cuchy problem consisting of the wve eqution together with the initil conditions u xx u yy =, u(x, ) = f(x), u y (x, ) =, for some known initil dtum f, is n exmple of well posed problem Indeed, working completely nlogously to Exmple 33, we cn see tht solution to the bove problem is given by u(x, y) = ( f(x y) + f(x + y) ) The proof of uniqueness of solution is more involved nd will be omitted (it is bsed on the so-clled energy property of the wve eqution) Finlly, to show the continuity of the solution to the initil dt, we consider lso the Cuchy problem ũ xx ũ yy =, together with the initil conditions ũ(x, ) = f(x), ũ y (x, ) =, ie, we consider different initil condition f for the Cuchy problem, giving new solution ũ Working s bove, we cn immeditely see tht the solution to this Cuchy problem is given by ũ(x, y) = ( f(x y) + f(x + y) ) Now, we look t the difference of the solutions of the two Cuchy problems bove We hve u(x, y) ũ(x, y) = ( ) ( ) ( (f(x y) ) ( ) f(x y)+f(x+y) f(x y)+ f(x+y) = f(x y) + f(x+y) f(x+y) ) Hence if the difference f(z) f(z) is smll for ll z R, then the difference u ũ will lso be smll! Tht is the solution depends continuously on the PDE coefficients nd the problem dt We now consider n exmple of n ill-posed problem, which is lso due to Hdmrd 4 Jcques Slomon Hdmrd ( ), French mthemticin 7

21 Exmple 37 The Cuchy problem consisting of the Lplce eqution u xx + u yy =, for π < x < π, nd y >, (ie, Ω = ( π/, π/) (, + )), together with the initil conditions for every n =, 3, 5,, nd u(x, ) =, u y (x, ) = e n cos(nx), for π x π u( π/, y) = = u(π/, y), for y As we shll see in Chpter, it is possible to clculte (using the method of seprtion of vribles) tht solution to the bove problem is given by n u(x, y) = e n cos(nx)sinh(ny) (It is esy to check tht this is solution to the Cuchy problem just by differentiting bck nd verifying tht it indeed stisfies the PDE nd the initil conditions) Now we study wht hppens s we vry the odd number n ppering in the initil conditions We cn see tht u y (x, ) = e n cos(nx) e n, ie, s we increse n, the initil condition u y (x, ) chnges t n exponentilly smll mnner Also, reclling the definition of the hyperbolic sine 5, we hve n n+ny u(x, y) = e e e n ny cos(nx)sinh(ny) = cos(nx) n n Notice tht when y, the exponent of the first exponentil is positive nd thus, chnge in n results to n exponentilly lrge chnge in u(x, y) Hence, smll chnge in the initil dt (relised when chnging the constnt n), result to exponentilly lrge chnge in the solution u for y! Hence the problem is ill-posed In Chpter, we shll consider pproprite conditions for ech type of liner second order equtions (elliptic, prbolic, hyperbolic), tht result to well-posed problems Problem 5 Find the solution to the Cuchy problem consisting of the wve eqution together with the initil conditions u xx u yy =, u(x, ) =, u y (x, ) = g(x), for some known initil dtum g Is this problem well-posed or ill-posed? Why? 5 We recll tht the hyperbolic sine nd the hyperbolic cosine re defined s sinh x := (ex e x ), nd cosh x := (ex + e x ) 8

22 Chpter Problems of Mthemticl Physics In this chpter we shll be concerned with the clssicl equtions of mthemticl physics, together with pproprite initil (nd boundry) conditions The Lplce eqution We begin the discussion with Lplce eqution: u =, for (x, x,,x d ) Ω R d, () where := ( ) xx +( ) xx + +( ) xd x d denotes the so-clled Lplce opertor in d dimensions; in prticulr, in two dimensions Lplce eqution reds: u = u xx + u yy =, for (x, y) Ω R, () where := ( ) xx + ( ) yy The non-homogeneous version of the Lplce eqution, nmely u = f in Ω (3) for some known function f : Ω R d R, is known s the Poisson eqution Lplce nd Poisson equtions model predomintely phenomen tht do not evolve in time, typiclly properties of mterils (elsticity, electric or grvittionl chrge), probbility densities of rndom vribles, etc As we sw in Chpter, Lplce (nd therefore, Poisson) eqution is of elliptic type in fct, Lplce eqution is the rchetypicl eqution of elliptic type (see lso Theorem 9 for the cnonicl form of PDEs of elliptic type) n Ω Ω u= Ω Ω u = Dirichlet b c u =f () Dirichlet boundry vlue problem N eum n n b c u n =f (b) Neumnn boundry vlue problem Figure : Dirichlet nd Neumnn boundry vlue problems For the problem to be well posed, we equip the Lplce eqution with conditions long the whole of the boundry Ω of the domin Ω We shll cll these boundry conditions We shll consider two types of In the previous chpter, we tlked bout the Cuchy problem consisting of PDE, together with initil conditions The term initil conditions is used for PDEs tht model evolution phenomen (ie, PDEs for which one vrible is time ), for which the Cuchy problem is well posed For elliptic PDEs, however, which model phenomen tht do not evolve in time, it is conventionl to use the term boundry conditions insted 9

23 boundry conditions, nmely the Dirichlet boundry condition: u(x, y) = f(x, y), for(x, y) Ω, where f : Ω R is known function, nd the Neumnn boundry condition: u (x, y) = f(x, y), n for(x, y) Ω, where u n (x, y) is the directionl derivtive of u in the direction of the unit outwrd norml vector n t the point (x, y) of the boundry Ω We shll refer to the Lplce eqution together with the Dirichlet boundry condition s the Dirichlet boundry vlue problem nd to the Lplce eqution together with the Neumnn boundry condition s the Neumnn boundry vlue problem (see Figure for n illustrtion) Next, we shll be concerned with finding the solution to the Lplce eqution with the bove boundry conditions

24 Seprtion of vribles For simplicity of the presenttion, let Ω = [, ] [, b] R be the rectngulr region with vertices the points (, ), (, ), (, b), nd (, b) We seek the (unique) solution u : Ω R to the Lplce boundry-vlue problem u = in Ω, (4) u(, y) = u(, y) = u(x, ) =, for x, y b, (5) u(x, b) = f(x), for x, (6) where f : [, ] R is known function We begin by mking the crucil ssumption tht the solution u of the problem (4) is of the form u(x, y) = X(x)Y (y), for some twice differentible functions of one vrible X nd Y (Indeed, if we find one solution to the problem (4), it hs to be necessrily the only solution, due to the uniqueness of the solution property described bove) Then we hve u xx = X (x)y (y) nd u yy = X(x)Y (y) Inserting this into the PDE u =, we rrive to X (x)y (y) + X(x)Y (y) =, or X (x) X(x) + Y (y) Y (y) =, fter division by X(x)Y (y), which we cn ssume to be non-zero without loss of generlity (for otherwise, the solution u is identiclly equl to zero which mens tht we found the solution if lso f =, or tht this is impossible if f ) This gives X (x) X(x) = Y (y) Y (y), (7) tht is, the left-hnd side depends only on the independent vrible x nd the right-hnd side depends only on the independent vrible y Since x nd y re independent vribles, the only possibility for the reltion (7) to hold is for both the left- nd the right-hnd sides to be constnt, sy equl to λ R From this we get the ordinry differentil equtions Now from the the boundry conditions we get X (x) λx(x) =, nd Y (y) + λy (y) = X()Y (y) = X()Y (y) = giving X() = X() =, nd X(x)Y () = giving Y () = Now we seprte 3 cses: whether λ is positive, negtive or zero The cse λ > : If λ >, then the two-point boundry vlue problem hs solution of the form X (x) λx(x) =, < x <, nd X() = X() =, X(x) = Acosh( λx) + B sinh( λx), for some constnts A, B R, which cn be determined using the boundry conditions X() = X() = We hve = X() = Acosh() = A, nd = X() = B sinh( λ), which implies tht lso B = This mens tht if λ >, we get X(x) = nd thus, the only solution is the trivil solution u(x, y) =, which is not cceptble s u(x, y) on the top boundry

25 The cse λ = : If λ =, then the two-point boundry vlue problem becomes it hs solution of the form X (x) =, < x <, nd X() = X() = ; X(x) = Ax + B, for some constnts A, B R, which cn be determined using the boundry conditions X() = X() = We hve then = X() = B, nd = X() = A, implying lso tht A = Hence if λ = we gin rrive to the trivil solution u = which is not cceptble The cse λ < : If λ <, then there exists κ R such tht λ = κ The two-point boundry vlue problem hs solution of the form X (x) + κ X(x) =, for < x <, nd X() = X() =, X(x) = Acos(κx) + B sin(κx), for some constnts A, B R, which cn be determined using the boundry conditions X() = X() = We hve = X() = Acos() = A, nd = X() = B sin(κ); this implies sin(κ) =, which mens κ = nπ for ny n =,, integers From this we find κ = nπ, nd, thus, we obtin the solutions X(x) = B n sin ( nπx), for ll n =,, integers nd B n R We now turn our ttention to Y, which stisfies the two-point boundry vlue problem Y (y) κ Y (y) =, for < y < b, nd Y () =, X(x)Y (b) = f(x) As before, the solution of the ODE is of the form Y (y) = C cosh(κy) + D sinh(κy), for some constnts C, D R, which cn be determined using the boundry conditions From the left boundry condition, we get = Y () = C cosh() = C, giving the fmily of solutions Y (y) = D n sinh(κy) = D n sinh ( nπy), for ll n =,, integers nd D n R Hence ll the functions of the form u n (x, y) := E n sin ( nπx ) (nπy) sinh, setting E n = B n D n for ll n =,,, re solutions to the Lplce problem (4) Since the Lplce eqution is liner, it is not hrd to see tht if two functions re solutions to Lplce eqution, then ny liner combintion of these functions is solution to the Lplce eqution lso Hence, we cn formlly write tht the solution of the Lplce problem (4) is of the form u(x, y) = n= E n sin ( nπx ) (nπy) sinh This is the so-clled Principle of Superposition, whereby the if two functions re solutions to liner homogeneous PDE then their liner combintion is lso solution (8)

26 (t this point the bove equlity is only forml, s we do not know if the bove series converges) Notice tht the E n s re still not determined; this is to be expected s we hve not yet mde use of the remining boundry condition u(x, b) = X(x)Y (b) = f(x) which, in view of (8) cn be written s f(x) = u(x, b) = n= E n sin ( nπx ) sinh (nπb ) = n= Ẽ n sin ( nπx), (9) where we hve set E n := E n sinh ( ) nπb To conclude the solution, we need to determine ll Ẽ n s such tht (9) is stisfied Describing how to do tht is the content of the next section Problem 6 Let Ω = [, ] [, b] R We seek the (unique) solution u : Ω R to the Lplce boundry-vlue problem u = in Ω, u(, y) = u(x, ) =, for x, y b, u x (, y) =, for y b, u x (x, b) = g(x), for x, where g : [, ] R is known function Using the method of seprtion of vribles, clculte the solution up to unknown constnts, nd give condition tht cn enble us to clculte these unknown constnts 3